Are you sure your AirPods Max have the latest firmware? This issue was addressed in an update right after the first version came out and people reported the issue you're describing:
If you set your AirPods Max down and leave them stationary for 5 minutes, they go into a low power mode to preserve battery charge. After 72 stationary hours out of the Smart Case, your AirPods Max go into a lower power mode that turns off Bluetooth and Find My to preserve battery charge further
[Archive link, as the latest Support doc doesn't have this wording any longer]
If you set your AirPods Max down and leave them stationary for 5 minutes, they go into a low power mode to preserve battery charge. After 72 stationary hours out of the Smart Case, your AirPods Max go into a lower power mode that turns off Bluetooth and Find My to preserve battery charge further
[Archive link, as the latest Support doc doesn't have this wording any longer]
[0]https://web.archive.org/web/20210315052229/https://support.a...